existence of nth root

Theorem.

If a∈R with a>0 and n is a positive integer, then there exists a unique positive real number u such that un=a.

Proof.

The statement is clearly true for n=1 (let u=a). Thus, it will be assumed that n>1.

Define p:ℝ→ℝ by p⁢(x)=xn-a. Note that a positive real root of p⁢(x) corresponds to a positive real number u such that un=a.

If a=1, then p⁢(1)=1n-1=0, in which case the existence of u has been established.

Note that p⁢(x) is a polynomial function and thus is continuous. If a<1, then p⁢(1)=1n-a>1-1=0. If a>1, then p⁢(a)=an-a=a⁢(an-1-1)>0. Note also that p⁢(0)=0n-a=-a<0. Thus, if a≠1, then the intermediate value theorem can be applied to yield the existence of u.